Groups

A group  is the least structure in which we can define an internal operation which generalizes the multiplication and division on nonzero real numbers. A field  is the nicest way in which two distinct groups can be fitted together so as to preserve the two identity elements as in the familiar example $(\mathbb{R} ,+,\times)$ which is given by $\bigl( (\mathbb{R} , + ), (\mathbb{R} \setminus \{ 0 \},\times)\bigr)$. A ring  is slightly weaker, lacking division, like $(\mathbb{Z} ,+,\times)$. A vector space, or linear space, is the nicest way in which a group (with a commutative operation +) can be combined with a field so as to preserve all three identity elements; a module  is similar, but uses a ring instead of a field for its scalars. For each of these, the appropriate maps which preserve the operations (hence all identities and inverses), between two structures of the same type, are called homomorphisms . Invertible homomorphisms are called isomorphisms and, for a given structure the set of self-isomorphisms or automorphisms forms a group. The fundamental concepts in group theory are enshrined in what we now call the category Grp of groups and group homomorphisms. Definitions
A group  is a set G together with a map

$$* : G \times G \ \to \ G : (g,h) \ \mapsto \ g * h,$$

called a binary operation , satisfying:

1.

* is associative: $(a * b) * c \ = \ a * (b * c)\ (\forall\, a,b,c \in G )$;

2.

* has an identity element $e \in G$: $a * e = e * a = a \ (\forall\, a \in G)$;

3.

* admits inverses: $(\forall\, a \in G)(\exists\, a^{-1} \in G): \ a * a^{-1} = a^{-1} * a = e$.

When there is no risk of confusion, we may omit the product symbol * and write ab for a*b; however, it is quite common to be dealing with more than one group structure on the same set so care is needed. A group (G,*) is called abelian or commutative if a*b = b * a for all $a,b \in G$. A map $\phi : G \to H$ between groups $(G,*), (H,\star)$ is a group homomorphism  if it preserves the group operations:

$$\phi (a * b) \ = \ \phi (a)\star \phi (b) \qquad (\forall\, a, b\in G)\, .$$

If the homomorphism is from a group to itself then we call it an endomorphism. A subset G' of a group G is a subgroup  of G if the inclusion map $G' \stackrel{i}{\hookrightarrow} G$ is a group homomorphism; then G' is itself a group with the restriction of the operation of G. The kernel  of a homomorphism $\phi : G \to H$ is the subgroup $\ker\phi=\phi^{\leftarrow}1_H$, where 1_H denotes the identity element of H. If a group homomorphism $\phi : G \to H$ is bijective, then its inverse is also a group homomorphism, $\phi$ is called an isomorphism and the groups G and H are called isomorphic, written $G \cong H.$ If an isomorphism is from a group to itself, then we call it an automorphism. If H is a subgroup of G, then we define for each $g \in G$:

1.

$gH = \{ g * h \mid h \in H\}$, and $\{gH\mid g\in G\}$ the set of right cosets of H in G.

2.

$Hg = \{h*g\mid h \in H\}$, and $\{Hg\mid g\in G\}$ the set of left cosets of H in G.

There is always a bijection between the sets of right and left cosets, but it may not be natural. When it is, we call the subgroup normal if gH = Hg for all g. In this case the set of cosets itself forms a group, the quotient group  denoted by G/H. The number of elements in G is called the order  of G, denoted |G|; if the order of a group is finite then we call it a finite group. If the smallest number of elements in a generating set is finite, then we call the group finitely generated. If |G| is finite and G has a subgroup H, then H has a finite number of right cosets in G, called the index  of H in G and denoted by (G:H). It follows that, if |G| is finite,

$$\vert G\vert = (G: H) \ \vert H\vert\, . \qquad (\mbox{Remember: } \vert G\vert \mbox{ finite!} )$$

This gives the famous theorem of Lagrange: if G is a finite group then the order of any subgroup divides the order of G. Hence groups of prime order have no nontrivial subgroups. We can construct a group from a given set of elements by simple juxtaposition of the elements; the group consists of the set of all finite words made up from the given elements and their inverses, with composition of words by juxtaposition. This group is called the free group  on the given elements. The free group on one generator is isomorphic to $(\mathbb{Z} , +)$; a free group on more than one generator cannot be abelian. Many groups arise in practice as a set of generating elements together with some rules of combination. The free product  G * F of two groups consists of words made from both, with all internal products simplified in each. The direct product  $(G \times H, *\times\circ)$ of two groups $(G,*), (H,\circ )$ is the group defined on the product set $G \times H$ by

$$(g,h) *\times\circ (g',h') = (g * g', h \circ h')\, .$$

If two normal subgroups J,K of a group G can be found such that every $g \in G$ can be written uniquely in the form g = jk for some $j \in J,\ k\in K$ and $J \cap K = \{ e\}$, the trivial subgroup, then we say that G decomposes into the direct product of J and K. The commutator of two elements a,b in a group G is the element a^-1b^-1ab. The subgroup [G,G] of G generated by all of its commutators is called the commutator subgroup; the quotient group G/[G,G] is always abelian. We call the process of taking this quotient abelianizing G. Exercises

1.

Show that the identity element in a group is unique, as are inverses.

2.

The set $\{ z \in \mathbb{C}\mid \ \vert z\vert = 1\}$, of unimodular complex numbers, forms an infinite group under multiplication. This is actually a topological group, homeomorphic to the unit circle.

3.

The set of n^th roots of unity forms a group under multiplication.

4.

Find all possible groups of orders 2, 3 and 4 by writing out possible entries in the matrix of products (the group table).

5.

Given a finite group G and any set $a_1,a_2,\ldots,a_n$ of distinct elements from G, prove that these elements and their products among themselves and their inverses generate a subgroup of G.

6.

The set of $n \times n$ nonsingular real matrices forms a group $GL (n,\mathbb{R} )$, often just written GL(n), the general linear group, under matrix multiplication. So does O(n), the subset consisting of orthogonal matrices, and its subset SO(n) consisting of those with determinant +1.

7.

Prove that GL(2) has a subgroup consisting of

$$\left\{ \left( \begin{array}{cc}\cos\theta & -\sin\theta\... ...ht) \left\vert \right. \ \theta\in \mathbb{R} \right\} .$$

This is actually SO(2), the special orthogonal group of $2\times 2$ real matrices.

8.

Find an isomorphism

$$f:SO(2)\rightarrow \{z\in\mathbb{C} \vert \ \vert z\vert=1\}$$

and give its inverse.

9.

Prove that, for all elements a in group G, the map

$$c_a:G\rightarrow G:x\mapsto a^{-1}xa$$

is an automorphism; find the inverse of c_a.

10.

Prove that if we have a homomorphism $f:G\rightarrow H$ and H is abelian, then $\ker f$ contains all of the commutators in G.

11.

If $\phi : G \longrightarrow H$ is a group homomorphism and e_H denotes the identity element in H, then:

(a)

$\ker \phi = \{ g \in G \mid \phi (g) = e_H\}$ is a subgroup of G.

(b)

${\rm im \ }\phi = \{ \phi (g) \in H \mid g \in G \}$ is a subgroup of H.

12.

The set of n^th roots of unity forms a subgroup of the abelian group of unimodular complex numbers.

13.

The map

$$\phi : \mathbb{Z}\longrightarrow \mathbb{S} ^1 : k \longmapsto e^{ik2\pi}$$

is a group homomorphism from the additive group of integers $(\mathbb{Z} , +)$ to the multiplicative group of unimodular complex numbers.

14.

$(\mathbb{Z} , +)$ is a subgroup of $(\mathbb{R} , + )$.

15.

$GL(n;\mathbb{R} )$ is not abelian if n > 1.

16.

The symmetric group S_n of permutations of n objects is not abelian for n > 2.

17.

Given groups G₁, G₂, show that the natural projections

$$p_i : G_1 \times G_2 \ \longrightarrow \ G_i : \ (g_1, g_2 ) \ \mapsto \ g_i \qquad (i = 1,2)$$

are group homomorphisms from the direct product group.

18.

If H,K are subgroups of G then $H \cap K$ is also a subgroup, but $H \cup K$ is a subgroup of G if and only if $H \subseteq K$ or $K \subseteq H$. If H,K are normal subgroups then so is HK.

19.

If G has no nontrivial subgroups (that is, only $\{e\}$ and G are subgroups of G) then G is generated by one element (so G is called a cyclic group) and has prime order.